Archimedean Choice Functions: an Axiomatic Foundation for Imprecise Decision Making
This work addresses foundational issues in imprecise decision-making for researchers in probability theory and decision analysis, but it is incremental as it builds on existing frameworks like choice functions and coherent lower previsions.
The paper tackles the problem of decision-making under imprecise probability models by providing axiomatic foundations for two decision rules, E-admissibility and maximality, using choice functions. It establishes representation theorems that uniquely characterize these rules in terms of coherent lower previsions.
If uncertainty is modelled by a probability measure, decisions are typically made by choosing the option with the highest expected utility. If an imprecise probability model is used instead, this decision rule can be generalised in several ways. We here focus on two such generalisations that apply to sets of probability measures: E-admissibility and maximality. Both of them can be regarded as special instances of so-called choice functions, a very general mathematical framework for decision making. For each of these two decision rules, we provide a set of necessary and sufficient conditions on choice functions that uniquely characterises this rule, thereby providing an axiomatic foundation for imprecise decision making with sets of probabilities. A representation theorem for Archimedean choice functions in terms of coherent lower previsions lies at the basis of both results.